On the numerical approximation of a problem involving a mixture of a MGT viscous material and an elastic solid

Abstract

In this work, we analyze, from the numerical point of view, a problem including a mixture made of a MGT viscoelastic solid and an elastic solid. The corresponding variational problem is a linear system composed of two coupled hyperbolic equations written in terms of the acceleration of the first constituent and the velocity of the second one. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one-dimensional numerical simulations are shown to demonstrate the accuracy of the proposed approximations and the behaviour of the solution.